Derivation

Split input into 2 regimes
if k < 6.493800852448035e+60
1. Initial program 0.1
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Applied simplify0.0
  \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}}\]
3. Using strategy rm
4. Applied div-inv0.0
  \[\leadsto \color{blue}{\left({k}^{m} \cdot a\right) \cdot \frac{1}{(\left(10 + k\right) \cdot k + 1)_*}}\]
if 6.493800852448035e+60 < k
1. Initial program 5.8
  \[\frac{a \cdot {k}^{m}}{\left(1 + 10 \cdot k\right) + k \cdot k}\]
2. Applied simplify5.8
  \[\leadsto \color{blue}{\frac{{k}^{m} \cdot a}{(\left(10 + k\right) \cdot k + 1)_*}}\]
3. Using strategy rm
4. Applied clear-num6.1
  \[\leadsto \color{blue}{\frac{1}{\frac{(\left(10 + k\right) \cdot k + 1)_*}{{k}^{m} \cdot a}}}\]
5. Taylor expanded around inf 6.1
  \[\leadsto \frac{1}{\color{blue}{10 \cdot \frac{k}{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}} + \left(\frac{1}{e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)} \cdot a} + \frac{{k}^{2}}{a \cdot e^{-1 \cdot \left(\log \left(\frac{1}{k}\right) \cdot m\right)}}\right)}}\]
6. Applied simplify0.6
  \[\leadsto \color{blue}{\frac{1}{(\left(\frac{k}{a}\right) \cdot \left({k}^{\left(-m\right)} \cdot \left(k + 10\right)\right) + \left(\frac{{k}^{\left(-m\right)}}{a}\right))_*}}\]
Recombined 2 regimes into one program.

Runtime

herbie shell --seed '#(1071725047 233389029 2036512464 3988615230 2972226563 1111574017)' +o rules:numerics
(FPCore (a k m)
  :name "Falkner and Boettcher, Appendix A"
  (/ (* a (pow k m)) (+ (+ 1 (* 10 k)) (* k k))))

Error

Derivation

`if k < 6.493800852448035e+60`

`if 6.493800852448035e+60 < k`

Runtime